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Operator Integrals, Spectral Shift, and Spectral Flow

Published online by Cambridge University Press:  20 November 2018

N. A. Azamov ,
A. L. Carey ,
P. G. Dodds  and
F. A. Sukochev
N. A. Azamov
Affiliation:
(Azamov, Dodds) School of Computer Science, Engineering and Mathematics, Flinders University of South Australia, Bedford Park, 5042, SA, Australia, azam0001@csem.flinders.edu.au, peter@csem.flinders.edu.au
P. G. Dodds
Affiliation:
(Carey)Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia, acarey@maths.anu.edu.au
F. A. Sukochev
Affiliation:
(Sukochev) School of Mathematics and Statistics, University of New South Wales, Kensington, NSW 2052, Australia, f.sukochev@unsw.edu.au
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Abstract

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We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman–Solomyak representation of the spectral shift function of M.G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.

Keywords

47A5647B4947A5546L51
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 2 , 01 April 2009 , pp. 241 - 263
Copyright
Copyright © Canadian Mathematical Society 2009

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